• The Grothendieck ring G0(H) is the abelian group generated by the iso. classes [V ] of finite dim’l (right) H-modules V modulo the relations [V ] = [U ] + [W ] for each s.e.s. 0 → U → V → W → 0. The iso. classes of simple modules form a Z-basis. Multiplication is given by ⊗ = ⊗k. • The character algebra R(H) is the k-subalgebra of H∗ that is generated by the image of the character map χ : G0(...

Using Penning-trap experiments and a shell-correction method incorporating ellipsoidal shape deformations, we investigate the formation and stability patterns of trianionic gold clusters. Theory and experiment are in remarkable agreement concerning appearance sizes and electronic shell effects. In contrast to multiply cationic clusters, decay of the trianionic gold clusters occurs primarily via...

where the LGLn(λ) are irreducible GLn(C)-modules and the S λ k are irreducible Sk-modules. The decomposition in (0.1) essentially makes the study of the representations of GLn(C) and the study of representations of the symmetric group Sk two sides of the same coin. The group GLn(C) has interesting subgroups, GLn(C) ⊇ On(C) ⊇ Sn ⊇ Sn−1, and corresponding centralizer algebras, CSk ⊆ CBk(n) ⊆ CAk(...

This paper is concerned with finite unions of ideals and modules. The first main result is that if N ⊆ N1 ∪N2 ∪ · · · ∪Ns is a covering of a module N by submodules Ni, such that all but two of the Ni are intersections of strongly irreducible modules, then N ⊆ Nk for some k. The special case when N is a multiplication module is considered. The second main result generalizes earlier results on co...

A core operation in actual circuits, especially in digital signal processing algorithms, is multiplication. Often, the computational performance of a DSP system is limited by its multiplication performance [Pet95]. The implementation of multiplier modules into FPGAs is crucial in terms of area, speed and pin limitation. In many cases, even small multiplier modules will exceed the capacity of on...

In this paper, all rings are associative with identity and all modules are unital left modules unless otherwise specified. Let R be a ring and M a module. N ≤M will mean N is a submodule of M. A submodule E of M is called essential in M (notation E ≤e M) if E∩A = 0 for any nonzero submodule A of M. Dually, a submodule S of M is called small in M (notation S M) if M = S+T for any proper submodul...

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Let R be a commutative ring with non-zero identity, S multiplicatively closed subset of and M unital R-module. In this paper, we define submodule N (N :R M) ? = to weakly S-primary if there exists s such that whenever m 0 , am N, then either sa ??(N or sm N. We present various properties characterizations concept (especially in faithful multiplication modules). Moreover, the behavior structure ...

The main goal of this paper is to introduce a new class in the category modules. It called quasi-invertibility monoform (briefly QI-monoform) This modules generalization Various properties and another characterization QI-monoform are investigated. So, we prove that an R-module M if only for each non-zero homomorphism f:M E(M), kernel not quasi-invertible submodule M. Moreover, cases under which...

The complex cellular network was formed by the interacting gene modules. Building 10 the high-quality RNA-seq-based Gene Co-expression Network (GCN) is critical for 11 uncovering these modules and understanding the phenotypes of an organism. Here, 12 we established and analyzed the RNA-seq-based GCNs in two monocot species rice 13 and maize, and two eudicot species Arabidopsis and soybean, and ...